Mathematical modeling of heating of a dimetallic plate by a high-energy concentrated radiation flux

Mathematical modeling of heat transfer processes proceeding together under conditions of intense phase transformations (melting and evaporation of metals) under laser action on a dimetallic target has been carried out. Based on the results of numerical simulation, it is established that the laser power exerts a significant influence on the dynamics of the melting process. It is shown that evaporation of metal in the boundary layer of the plate can also have a significant effect on the heat transfer characteristics. It is established that metal vapors forming a gas mixture in the near-wall region can absorb laser radiation and cause the phenomenon of optical resonance with increasing their critical concentration, in other words, when the metal melts, optical breakdown is possible.


Introduction
Laser treatment of metals in industry is becoming increasingly important due to its advantages such as high processing accuracy, the absence of mechanical damage on the processed products, and as a result, a lower consumption of metal for tolerances and allowances of blanks, as well as the ability to work with complex geometric shapes [1,2].It should be noted that industrial laser metal processing plants are less harmful for the environment, since they do not generate waste industrial oils that are used to cool and lubricate moving parts of mechanical metalworking machines.
In connection with the growing interest in laser processing of metal, studies of the processes that occur during laser processing become topical, however, field experiments are quite complex and expensive.In these cases, the most suitable is the apparatus of mathematical modeling, which makes it possible to carry out studies of the processes under consideration with high accuracy and efficiency without any material costs.
The melting process is accompanied by a change in the structure of the material with a change in the thermophysical characteristics: heat capacity, density, thermal conductivity.It should be noted that under the conditions of laser action, intense heating of the metal takes place in a limited region.Accordingly, in the plate structure it is possible to form an essentially nonlinear and spatially-distributed fusion front, the shape of which does not yield to polynomial approximation.

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A significant contribution to the solution of the multidimensional formulation of Stefan problem was made by V.I.Mazhukin [3][4][5], A.A. Samarskii [6,7], and L.I. Rubinshtein [8].However, the methods of [3][4][5][6][7][8] are based on the explicit separation of the fusion front, and this requires the use of a very complex mathematical apparatus based on a combination of the experimental results of the melting process and the apparatus of differential topology that takes into account the dynamic changes in the calculated difference grids.
The aim of the work is mathematical modeling of the processes of heat and mass transfer proceeding together under conditions of intense phase transformations (melting and evaporation of a metal) under the action of a laser.

Physical statement of the problem
At the initial instant of time (t = 0), a laser beam is incident on the bounded region of the dimetallic target, as a result, the metal of both layers begins to heat up.When the phase transition temperatures are reached, the process of metal melting is initiated, resulting in a heterogeneous structure consisting of liquid and solid phases of metals with different thermophysical characteristics.With continued heating, the process of evaporation of the metal into the gas region is initiated.The metal vapor forms a vapor-gas mixture in the external environment, which has a significant effect on the radiation transfer characteristics.
When solving the problem, it was assumed that the thermophysical characteristics of metals depend on the position of the fusion front.The thermophysical characteristics of metals and gas, the heat of melting and evaporation of metals are given in Table 1.The laser beam is directed perpendicular to the target surface.
Table 1.Basic physical characteristics in the system "Dimetallic plate-gas".

Thermophysical characteristics of metals Solid phase
Fluid phase c, J/(kg•К)

Mathematical formulation of the problem
The mathematical model of the process under consideration includes the energy equations for metals in dimensionless form, taking into account the dependence of thermophysical characteristics on temperature: The energy equation for the upper plate: The energy equation for the lower plate: Heat and mass transfer in the thermal control system of technical and technological energy equipment here C = c / c0, P = ρ / ρ0, Λ = λ / λ0, θ = T / T0 are the dimensionless heat capacity, density, thermal conductivity, and temperature, respectively; c0, ρ0, λ0, T0 are the dimensionless parameters X = x / l, Y = y / l are dimensionless coordinates; Pom1, Pom2 -Pomerantsev criteria for the melting process for the upper and lower plates, respectively; Fo1, Fo2 -Fourier numbers for the upper and lower plates, respectively; W1, W2 -melting rates of plates, kg / (m 2 • s); Q1, Q2 -heat of melting of metals, kJ / kg; δ is the Dirac delta function; h, l -linear dimensions of plates, m.The energy equation for the described process in dimensionless quantities for a gas: here I0 is the intensity of the laser radiation, W/m 2 ; Eisl = I/I0 is the dimensionless radiation intensity.
The diffusion equation in the gas region: here C is the mass concentration of metal vapor in the gas, kg/kg; Fod is the Fourier diffusion number.The diffusion coefficient was determined according to [11]: here σ12 is the parameter of the function expressing the potential interaction energy of the molecules of the first and second components; Ω (1,1) * (Т * ) is the reduced interaction integral of the molecules of the first and second components; M1, M2 are molar masses, kg / kmol; P is the pressure, Pa.The reduced interaction integral of the molecules of the first and second components [11]: here J1, J2 is the ionization energy of the metal atom and the gas molecule, eV; α is the gas polarizability coefficient; k = 0.863•10 -4 eV / K is the Boltzmann constant; m is the temperature dependence of D 12 ∼T n ; a, b are coefficients that depend on the values of J1.The radiation transfer equation [12]: here Ω is the direction of the ray; I -radiation intensity, W / m 2 ; χ is the absorption coefficient of the medium, m -1 ; σ is the Stefan-Boltzmann constant, W•m -2 •K -4 .The physical constants for equations (5)(6)(7) are given in [8,10,13].The system of equations (1-7) was solved under the following boundary conditions (Fig. 1 , t > 0. The problem was solved by the method of finite differences using the implicit four-point difference template with the use of discontinuous coefficients [13,14].The nonstationary multidimensional Stefan problem was solved by the method of implicit separation of the fusion front [14].

Results
When solving the problem, a dimetallic plate was considered, the upper layer of which was made of aluminum, and the lower one was made of lead.The choice of these materials is due to their extensive use in industry.With increasing intensity of the incident radiation, the concentration of metal vapor in the gas region increases to a critical value, and then begins to decrease.This can be explained by the intensive outflow of metal vapors from the region under consideration, owing to a greatly increased diffusion coefficient.According to (5), the diffusion coefficient depends nonlinearly on the temperature.With a continuous increase in the radiation intensity, starting from a certain instant of time, the diffusion coefficient becomes so large that an intensive outflow of metal vapors occurs and the concentration decreases.According to the results of mathematical modeling, it is established that with sufficiently intense laser action (> 900 MW/m 2 ), the effect of optical breakdown in the near-wall gas region is possible.This is due to the fact that metal vapors, forming a gas mixture in the nearwall region, can substantially absorb laser radiation, as a result, an intense, even exponential temperature increase is possible.The latter leads to an increase in the optical thickness of the wall gas layer, and also forms zones with intense self-heating in the gas region as a result of the phenomenon of reradiation.In other words, in some cases of metal melting, there are risks of destruction of the metal preform and damage to the laser.
As can be seen from Fig. 3,a, melting of the lower plate can begin even before the moment of contact with it of the melting front of the upper plate.

Conclusion
It is established that the laser power exerts a significant influence on the dynamics of the melting process.It is shown that the evaporation of metal into the near-surface gas layer can MATEC Web of Conferences 194, 01046 (2018) https://doi.org/10.1051/matecconf/201819401046HMTTSC-2018 Matec Web of Conferences also have a significant effect on the heat transfer characteristics.It is established that metal vapors, forming a gas mixture in the near-wall region, can substantially absorb laser radiation and cause the phenomenon of optical resonance.
In conclusion, we can say that in order to ensure a fairly stable process of melting by a laser beam, it is necessary to organize a constant flow of inert gas in the melting and welding region, which ensures the entrainment of metal vapors from the melting region.

Fig. 2 (Fig. 2 .
Fig. 2. Dependence of temperature (a) and metal vapor concentration (b) in the gas on the intensity of the incident radiation at the upper plate-gas interface in the middle of the upper plate.